The articles in this book are based on talks given at the international conference ""Lie algebras, vertex operator algebras and their applications"", in honor of James Lepowsky and Robert Wilson on their sixtieth birthdays, held in May of 2005 at North Carolina State University. Some of the papers in this volume give inspiring expositions on the development and status of their respective research areas. Others outline and explore the challenges as well as the future directions of research for the twenty-first century. The focus of the papers in this volume is mainly on Lie algebras, quantum groups, vertex operator algebras and their applications to number theory, combinatorics and conformal field theory. This book is useful for graduate students and researchers in mathematics and mathematical physics who want to be introduced to different areas of current research or explore the frontiers of research in the areas mentioned above.
Lie algebras and related topics: A class of gradings of simple Lie algebras by K. Baur and N. Wallach Conjugacy results for the Lie algebra $\mathfrak{sl}_2$ over an algebra which is a UFD by S. Berman and J. Morita Kirillov-Reshetikhin modules associated to $G_2$ by V. Chari and A. Moura Support spaces and Auslander-Reiten components by R. Farnsteiner On the cohomology of modular Lie algebras by J. Feldvoss Eisenstein series on loop groups: Maass-Selberg relations 4 by H. Garland Generalized Littlewood-Richardson rule for exceptional Lie algebras $E_6$ and $F_4$ by A. Hoshino An introduction to $Q_n$ and its graph related quotients by D. Nacin Affine geometric crystal of type $G^{(1)}_2$ by T. Nakashima New constructions of Yang-Baxter systems by F. F. Nichita and D. Parashar Construction of some algebras associated to directed graphs and related to factorizations of noncommutative polynomials by V. Retakh, S. Serconek, and R. L. Wilson Geometric and combinatorial realizations of crystals of enveloping algebras by A. Savage Lie algebras of small dimension by H. Strade Vertex (operator) algebras and related topics: Symmetric polynomials and $H_D$-quantum vertex algebras by I. I. Anguelova $H_T$-vertex algebras by M. J. Bergvelt On intertwining operators and recursions by C. Calinescu Representations of vertex operator algebras by C. Dong and C. Jiang On nonsemisimple fusion rules and tensor categories by J. Fuchs The duality between vertex operator algebras and coalgebras, modules and comodules by K. Hubbard Some developments in vertex operator algebra theory, old and new by J. Lepowsky Twisted modules and quasi-modules for vertex operator algebras by H. Li Chiral algebras and partition functions by G. Mason and M. P. Tuite Modular forms and almost linear dependence of graded dimensions by A. Milas $(k,r)$-Admissible configurations and intertwining operators by M. Primc Hilbert schemes of points on the minimal resolution and soliton equations by Z. Qin and W. Wang Twining characters and Picard groups in rational conformal field theory by C. Schweigert, J. Fuchs, and I. Runkel.
